| Step | Hyp | Ref
| Expression |
|---|
| 1 | | neips.1 |
. . . . . . . 8
β’ π = βͺ
π½ |
| 2 | 1 | clsss3 22562 |
. . . . . . 7
β’ ((π½ β Top β§ π β π) β ((clsβπ½)βπ) β π) |
| 3 | 2 | sseld 3981 |
. . . . . 6
β’ ((π½ β Top β§ π β π) β (π β ((clsβπ½)βπ) β π β π)) |
| 4 | 3 | impr 455 |
. . . . 5
β’ ((π½ β Top β§ (π β π β§ π β ((clsβπ½)βπ))) β π β π) |
| 5 | 1 | isneip 22608 |
. . . . 5
β’ ((π½ β Top β§ π β π) β (π β ((neiβπ½)β{π}) β (π β π β§ βπ β π½ (π β π β§ π β π)))) |
| 6 | 4, 5 | syldan 591 |
. . . 4
β’ ((π½ β Top β§ (π β π β§ π β ((clsβπ½)βπ))) β (π β ((neiβπ½)β{π}) β (π β π β§ βπ β π½ (π β π β§ π β π)))) |
| 7 | | 3anass 1095 |
. . . . . . . . . . 11
β’ ((π½ β Top β§ π β π β§ π β ((clsβπ½)βπ)) β (π½ β Top β§ (π β π β§ π β ((clsβπ½)βπ)))) |
| 8 | 1 | clsndisj 22578 |
. . . . . . . . . . 11
β’ (((π½ β Top β§ π β π β§ π β ((clsβπ½)βπ)) β§ (π β π½ β§ π β π)) β (π β© π) β β
) |
| 9 | 7, 8 | sylanbr 582 |
. . . . . . . . . 10
β’ (((π½ β Top β§ (π β π β§ π β ((clsβπ½)βπ))) β§ (π β π½ β§ π β π)) β (π β© π) β β
) |
| 10 | 9 | anassrs 468 |
. . . . . . . . 9
β’ ((((π½ β Top β§ (π β π β§ π β ((clsβπ½)βπ))) β§ π β π½) β§ π β π) β (π β© π) β β
) |
| 11 | 10 | adantllr 717 |
. . . . . . . 8
β’
(((((π½ β Top
β§ (π β π β§ π β ((clsβπ½)βπ))) β§ π β π) β§ π β π½) β§ π β π) β (π β© π) β β
) |
| 12 | 11 | adantrr 715 |
. . . . . . 7
β’
(((((π½ β Top
β§ (π β π β§ π β ((clsβπ½)βπ))) β§ π β π) β§ π β π½) β§ (π β π β§ π β π)) β (π β© π) β β
) |
| 13 | | ssdisj 4459 |
. . . . . . . . . 10
β’ ((π β π β§ (π β© π) = β
) β (π β© π) = β
) |
| 14 | 13 | ex 413 |
. . . . . . . . 9
β’ (π β π β ((π β© π) = β
β (π β© π) = β
)) |
| 15 | 14 | necon3d 2961 |
. . . . . . . 8
β’ (π β π β ((π β© π) β β
β (π β© π) β β
)) |
| 16 | 15 | ad2antll 727 |
. . . . . . 7
β’
(((((π½ β Top
β§ (π β π β§ π β ((clsβπ½)βπ))) β§ π β π) β§ π β π½) β§ (π β π β§ π β π)) β ((π β© π) β β
β (π β© π) β β
)) |
| 17 | 12, 16 | mpd 15 |
. . . . . 6
β’
(((((π½ β Top
β§ (π β π β§ π β ((clsβπ½)βπ))) β§ π β π) β§ π β π½) β§ (π β π β§ π β π)) β (π β© π) β β
) |
| 18 | 17 | rexlimdva2 3157 |
. . . . 5
β’ (((π½ β Top β§ (π β π β§ π β ((clsβπ½)βπ))) β§ π β π) β (βπ β π½ (π β π β§ π β π) β (π β© π) β β
)) |
| 19 | 18 | expimpd 454 |
. . . 4
β’ ((π½ β Top β§ (π β π β§ π β ((clsβπ½)βπ))) β ((π β π β§ βπ β π½ (π β π β§ π β π)) β (π β© π) β β
)) |
| 20 | 6, 19 | sylbid 239 |
. . 3
β’ ((π½ β Top β§ (π β π β§ π β ((clsβπ½)βπ))) β (π β ((neiβπ½)β{π}) β (π β© π) β β
)) |
| 21 | 20 | exp32 421 |
. 2
β’ (π½ β Top β (π β π β (π β ((clsβπ½)βπ) β (π β ((neiβπ½)β{π}) β (π β© π) β β
)))) |
| 22 | 21 | imp43 428 |
1
β’ (((π½ β Top β§ π β π) β§ (π β ((clsβπ½)βπ) β§ π β ((neiβπ½)β{π}))) β (π β© π) β β
) |
